Existence and multiplicity of solutions for fractional $p_{1}(x,\cdot )\& p_{2}(x,\cdot )$-Laplacian Schrödinger-type equations with Robin boundary conditions

Abstract

In this paper, we study fractional $p_{1}(x,\cdot )\& p_{2}(x,\cdot )$-Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland’s variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in $\mathbb{R}^{N}\setminus \overline{\Omega}$ for fractional order $p_{1}(x,\cdot )\& p_{2}(x,\cdot )$-Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results.

1 Introduction and the main results

In this paper, we consider fractional $p_{1}(x,\cdot )\& p_{2}(x,\cdot )$-Laplacian Schrödinger-type equations, including concave-convex nonlinearities with nonlocal Robin boundary conditions

$$ \textstyle\begin{cases} \sum_{i=1} ^{2} [(-\Delta )^{s}_{p_{i}(x, \cdot )}\varphi +V_{i}(x) \vert \varphi \vert ^{\bar{p}_{i}(x)-2}\varphi ] \\ \quad =\lambda _{1}A_{1}(x) \vert \varphi \vert ^{r_{1}(x)-2}\varphi +\lambda _{2}A_{2}(x) \vert \varphi \vert ^{r_{2}(x)-2}\varphi , \quad x \in \Omega , \\ \sum_{i=1} ^{2} [\mathcal{N}_{s,p_{i}(x, \cdot )}\varphi +\beta (x) \vert \varphi \vert ^{\bar{p}_{i}(x)-2}\varphi ] =\sum_{i=1} ^{2}g_{i}(x), \quad x \in \mathbb{R}^{N} \setminus \overline{\Omega}, \end{cases} $$

(1)

where $V_{i}(x)$ ($x\in \Omega$, $i=1,2$) is a potential function, $\Omega \subset \mathbb{R}^{N}$ ($N\geq 2$) is a bounded domain with the Lipschitz boundary ∂Ω, $s\in (0,1)$, $p_{i}(x,\cdot ):\mathbb{R}^{2N}\rightarrow (1,+\infty )$, $\overline{p}_{i}(x)=p_{i}(x,x)$, $r_{1}(x)$, $r_{2}(x)$ are continuous functions, $\lambda _{1}$, $\lambda _{2}$ are positive constants, $A_{1}(x)$, $A_{2}(x)$ are positive weighted functions, $g_{i}(x)\geq 0\in L^{1}(\mathbb{R}^{N}\setminus \Omega )$, $\beta (x)\geq 0\in L^{\infty}(\mathbb{R}^{N}\setminus \Omega )$,

$$ \mathcal{N}_{s,p_{i}(x,\cdot )}\varphi (x)= \int _{ \Omega} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{i}(x,y)-2}}(\varphi (x)-\varphi (y))}{ \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dy,\quad x\in \mathbb{R}^{N} \setminus \overline{\Omega}, $$

and

$$ (-\Delta )_{p_{i}(x,\cdot )}^{s}\varphi (x):=P.V. \int _{\mathbb{R}^{N}} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{i}(x,y)-2}}(\varphi (x)-\varphi (y))}{ \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dy, \quad x\in \Omega , $$

where $P.V$. stands for the Cauchy principal value.

Equations (1) arise from general reaction-diffusion equation

$$ \varphi _{t}=\nabla \cdot \bigl[A(\varphi )\nabla \varphi \bigr]+r(x,\varphi ), $$

(2)

where $A(\varphi )=|\nabla \varphi |^{p-2}+|\nabla \varphi |^{q-2}$. Problem (2) has applications in biophysics, plasma physics, and chemical reactions. For more details on equation (2), readers are referred to [1, 2]. Combining with a $\mathbb{Z}_{2}$-symmetric version of the mountain pass lemma for even functionals and some adequate variational methods, Mihăilescu [3] proved that the equations

$$ \textstyle\begin{cases} -\operatorname{div}(( \vert \nabla \varphi \vert ^{p_{1}(x)}+ \vert \nabla \varphi \vert ^{p_{2}(x)}) \nabla \varphi ) =f(x,\varphi ), \quad x\in \Omega , \\ \varphi =0,\quad x\in \partial \Omega \end{cases} $$

(3)

have infinitely many weak solutions. In addition, Chung and Toan [4] considered a class of fractional Laplacian problems

$$ \textstyle\begin{cases} (\Delta )_{p_{1}(x,\cdot )}^{s}\varphi (x)+(\Delta )_{p_{2}(x, \cdot )}^{s}\varphi (x) + \vert \varphi \vert ^{q(x)-2}\varphi \\ \quad =\lambda A_{1}(x) \vert \varphi (x) \vert ^{r_{1}(x)-2}\varphi (x) -\mu A_{2}(x) \vert \varphi (x) \vert ^{r_{2}(x)-2}\varphi (x), \quad x\in \Omega , \\ \varphi (x)=0, \quad x\in \partial \Omega \end{cases} $$

(4)

using variational techniques and Ekeland’s variational principle. The authors used the variational techniques to discuss the results of the existence of solutions in fractional cases [5–7]. In addition, Heidarkhani et al. [8–10] studied the existence results of variable exponent equations using variational methods and established the critical point theory. Zuo et al. [11] investigated the existence and multiplicity of solutions for the $p(x,\cdot )\& q(x,\cdot )$ fractional Choquard problems with variable order. On a similar issue, a related study was conducted by Biswas et al. For more details, see [12].

The classical Schrödinger equation is of the following form:

$$ ih\frac{\partial}{\partial t}\varphi =-\frac{h^{2}}{2m}\nabla ^{2} \varphi +V \varphi , $$

where V, φ denote the potential function and wave function, respectively, and i, h are constants ([13]). Recently, Xiang et al. in [14] and Bu et al. in [15] discussed the fractional Laplace operator Schrödinger equations with variable order and Schrödinger–Kirchhof-type equations, respectively.

The critical local problem involving concave-convex nonlinearities was first studied by Ambrosetti et al. in [16]. Subsequently, variational methods were used [17] to discuss the following equations:

$$ \textstyle\begin{cases} -\operatorname{div}(\omega (x) \vert \nabla \varphi \vert ^{p(x)-2}\nabla \varphi ) = \lambda a(x) \vert \varphi \vert ^{q(x)-2}\varphi +\mu b(x) \vert \varphi \vert ^{h(x)-2} \varphi , \quad x\in \Omega , \\ \varphi =0, \quad x\in \partial \Omega \end{cases} $$

(5)

with the variable order concave-convex term. For other similar types of equations, see [18, 19] and the references therein.

The Robin and Neumann boundary problems are interesting topics [20]. Mugnai et al. [21] investigated fractional p-Laplacian problems with nonlocal Neumann boundary conditions. Moreover, Deng [22] considered the following equations:

$$ \textstyle\begin{cases} -\Delta _{p(x)} \varphi =\lambda f(x,\varphi ),\quad x\in \Omega , \\ \vert \nabla \varphi \vert ^{p(x)-2}\frac{\partial \varphi}{\partial \eta}+\beta (x) \vert \varphi \vert ^{p(x)-2} \varphi =0, \quad x\in \partial \Omega . \end{cases} $$

(6)

For double-phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian, Manouni et al. [23] proved the existence of solutions by variational tools, truncation techniques, and comparison methods. In many papers, the Robin and Neumann boundary problems of fractional equations were studied in different ways; e.g., the Morse theory was used in [24, 25], the mountain pass lemma in [26, 27], Ekeland’s variational principle in [28, 29], and the topological degree in [30].

To our knowledge, there is no previous work on the problem (1). This paper is devoted to this topic. We obtain new results by applying the mountain pass lemma, Ekeland’s variational principle, the fountain theorem, and the Krasnoselskii genus theory. Our problem differs from problems (3), (4), and (5) in that we discuss Robin boundary conditions, and it also differs from problem the (6) in that we consider $p_{1}(x,\cdot )\& p_{2}(x,\cdot )$-Laplacian Schrödinger-type equations with concave-convex nonlinearities.

Before stating the main results, we introduce the basic assumptions.

• (P) $p_{i}(x,y)$ is a symmetric and continuous function, that is,

  $$ p_{i}(x,y)=p_{i}(y,x), \quad \text{for all } (x, y)\in \mathbb{R}^{N} \times \mathbb{R}^{N} $$

  with

  $$ 1< p_{i}^{-}:=\min_{(x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N}}p_{i}(x,y) \leq p_{i}(x,y)\leq p_{i}^{+}:=\max _{(x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N}}p_{i}(x,y)< +\infty , $$

  and

  $$ 1< p_{1}^{-}\leq p_{1}^{+}\leq p_{2}^{-}\leq p_{2}^{+}< +\infty , $$

  such that $sp_{i}^{+}< N$. Let $0< s<1<p(x,\cdot )$, the fractional critical exponent $p^{*}_{s}(x)$ be defined as $p^{*}_{s}(x)=\frac{Np(x,x)}{N-sp(x,x)}$ and $p(x,\cdot )< p^{*}_{s}(x)$ for all $x\in \overline{\Omega}$.

• (G) $\int g_{i}(x)\varphi \,dx=-\int g_{i}(x)\varphi \,dx$.

• (V) $V_{i}(x)$ is a continuous function, satisfying $\inf_{x\in \Omega}V_{i}(x)>V_{i0}>0$, for all $d_{i}>0$, $\mathrm{means} (\{x \in \Omega :V_{i}(x)< d_{i}\})<+\infty$.

• (H) $A_{1}(x)$ and $A_{2}(x)$ are weighted functions in $C(\overline{\Omega})$ and satisfy $A_{1}(x)\in L^{s_{1}(x)}(\Omega )$ such that $1< s_{1}(x) \in C(\overline{\Omega})$ and $1< s'_{1}(x)r_{1}(x)< p^{*}_{s}(x)$ for all $x\in \overline{\Omega}$, $A_{2}(x)\in L^{s_{2}(x)}(\Omega )$ such that $1< s_{2}(x) \in C(\overline{\Omega})$ and $1< s'_{2}(x)r_{2}(x)< p^{*}_{s}(x)$ for all $x\in \overline{\Omega}$. Here, $s'_{1}(x)$ and $s'_{2}(x)$ are conjugate exponents of the functions $s_{1}(x)$ and $s_{2}(x)$, respectively.

The main results of this paper are as follows:

Theorem 1.1

Assume that assumptions (P), (G), (V), and (H) hold. Equations (1) have two nontrivial weak solutions.

Theorem 1.2

Assume that assumptions (P), (G), (V), and (H) hold. Then, equations (1) have infinitely many nontrivial weak solutions in X.

Theorem 1.3

Assume that assumptions (P), (G), (V), and (H) hold. Then, equations (1) possess infinitely many solutions.

In Sect. 2, we state some basic results of the Lebesgue space $L^{q(x)}(\Omega )$. In Sect. 3, we introduce the workspaces associated with equations (1). In Sect. 4, we verify the $(PS)$ conditions and prove Theorem 1.1 by the mountain pass lemma and Ekeland’s variational principle. In Sect. 5, we prove Theorem 1.2 by applying the fountain theorem. Finally, using the Krasnoselskii genus theory, we give the proof of Theorem 1.3.

2 Preliminaries

In this section, we recall some basic results of the Lebesgue space $L^{q(x)}(\Omega )$ with a variable exponent. Assume that domain Ω is bounded in $\mathbb{R}^{N}$ with the Lipschitz boundary ∂Ω. Let

$$ q^{-}=\min_{x\in \overline{\Omega}}q(x), \qquad q^{+}=\max _{x\in \overline{\Omega}}q(x), $$

where $C_{+}(\overline{\Omega})= \{q\in C(\overline{\Omega}): q(x)>1, \text{for all } x\in \overline{\Omega} \}$.

The variable exponent Lebesgue space $L^{q(x)}(\Omega )$, which is defined by

$$ L^{q(x)}(\Omega )=\biggl\{ \varphi | \varphi :\Omega \rightarrow \mathbb{R} \text{ is measurable and } \int _{\Omega} \bigl\vert \varphi (x) \bigr\vert ^{q(x)}\,dx< \infty \biggr\} , $$

equipped with the Luxemburg norm

$$ \Vert \varphi \Vert _{q(x)}=\inf \biggl\{ \iota >0: \int _{\Omega} \biggl\vert \frac{\varphi (x)^{q(x)}}{\iota} \biggr\vert \,dx \leq 1 \biggr\} , $$

where $(L^{q(x)}(\Omega ), \|\cdot \|_{q(x)})$ is a separable, uniformly convex, and reflexive Banach space [31].

Let $L^{q'(x)}(\Omega )$ be the conjugate space of $L^{q(x)}(\Omega )$ and $1/q(x)+1/q'(x)=1$ ($p(x)$ and $q'(x)$ are conjugate indices to each other). For $\varphi \in L^{q(x)}(\Omega )$ and $v\in L^{q'(x)}(\Omega )$, the Hölder inequality

$$ \biggl\vert \int _{\Omega}\varphi (x)v(x)\,dx \biggr\vert < \biggl( \frac{1}{q^{-}}+ \frac{1}{q^{\prime -}} \biggr) \Vert \varphi \Vert _{q(x)} \Vert v \Vert _{q'(x)}\leq 2 \Vert \varphi \Vert _{q(x)} \Vert v \Vert _{q'(x)} $$

(7)

holds. If $q_{i}(x)\in C_{+}(\overline{\Omega})$ ($i=1,2,\ldots ,\widehat{n}$) and

$$ \frac{1}{q_{1}(x)}+ \frac{1}{q_{2}(x)}+\cdots + \frac{1}{q_{\widehat{n}}(x)}=1, $$

for all $\varphi _{i}(x)\in L^{q_{i}(x)}(\Omega )$, there exists

$$ \biggl\vert \int _{\Omega}\varphi _{1}(x)\varphi _{2}(x) \cdots \varphi _{ \widehat{n}}(x)\,dx \biggr\vert \leq \biggl( \frac{1}{q_{1}^{-}}+ \frac{1}{q_{2}^{-}}+\cdots \frac{1}{q_{\widehat{n}}^{-}} \biggr) \Vert \varphi _{1} \Vert _{q_{1}(x)} \Vert \varphi _{2} \Vert _{q_{2}(x)}\cdots \Vert \varphi _{ \widehat{n}} \Vert _{q_{\widehat{n}}(x)}. $$

Lemma 2.1

([32]) Let $\rho _{q(x)}$ be the modular of the $L^{q(x)}(\Omega )$ space, and $\rho _{q(x)}: L^{q(x)}(\Omega )\rightarrow \mathbb{R}$ defined by $\rho _{q(x)}(\varphi )=\int _{\Omega}|\varphi (x)|^{q(x)}\,dx$. Then, the following properties hold:

• (i) $\|\varphi \|_{q(x)}<1(=1, >1) \Longleftrightarrow \rho _{q(x)}( \varphi )<1 (=1, >1)$;

• (ii) ${\parallel \phi \parallel }_{q(x)}>1⟹{\parallel \phi \parallel }_{q(x)}^{q^{-}}\le {\rho }_{q(x)}(\phi )\le {\parallel \phi \parallel }_{q(x)}^{q^{+}}$;

• (iii) ${\parallel \phi \parallel }_{q(x)}<1⟹{\parallel \phi \parallel }_{q(x)}^{q^{+}}\le {\rho }_{q(x)}(\phi )\le {\parallel \phi \parallel }_{q(x)}^{q^{-}}$.

Lemma 2.2

([32]) If $\varphi , \varphi _{n}\in L^{q(x)}(\Omega )$ with $n\in \mathbb{N}$, then

• (i) $\lim_{n\rightarrow +\infty}\|\varphi _{n}-\varphi \|_{q(x)}=0$;

• (ii) $\lim_{n\rightarrow +\infty}\rho _{q(x)}(\varphi _{n}-\varphi )=0$;

• (iii) $\varphi _{n}(x)\rightarrow \varphi (x)$ a. e. in Ω and $\lim_{n\rightarrow +\infty}\rho _{q(x)}(\varphi _{n})=\rho _{q(x)}( \varphi )$.

Lemma 2.3

([33]) Let $p(x)$, $q(x)$ be measurable functions such that $p(x)\in L^{\infty}(\mathbb{R}^{N})$ and $1< p(x)q(x)<\infty$, for any $x\in \mathbb{R}^{N}$. Then, there is

$$ \min \bigl\{ \Vert \varphi \Vert ^{p^{+}}_{p(x)q(x)}, \Vert \varphi \Vert ^{p^{-}}_{p(x)q(x)} \bigr\} \leq \bigl\Vert \vert \varphi \vert ^{p(x)} \bigr\Vert _{q(x)} \leq \max \bigl\{ \Vert \varphi \Vert ^{p^{+}}_{p(x)q(x)}, \Vert \varphi \Vert ^{p^{-}}_{p(x)q(x)}\bigr\} $$

with $\varphi \in L^{q(x)}(\mathbb{R}^{N})$, $\varphi \neq 0$.

3 The basic properties of functionals and operators

In this section, we state some properties of functionals and operators, and give the definition of weak solutions of equations (1) with Robin boundary conditions. We first introduce the workspaces $(W,\|\cdot \|_{W})$ and $(X,\|\cdot \|_{X})$ associated with equations (1).

The fractional variable Sobolev space $W:=W^{s,q(x),p(x,\cdot )}(\Omega )$ is given by

$$\begin{aligned} W^{s,q(x),p(x,\cdot )}(\Omega ):={}& \biggl\{ \varphi : \Omega \rightarrow \mathbb{R} | \varphi \in L^{q(x)}(\Omega ), \\ &{} \int _{\Omega \times \Omega} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p(x,y)}}}{\mu ^{p(x,y)} \vert x-y \vert ^{N+sp(x,y)}}\,dx\,dy< \infty , \text{for some } \mu >0 \biggr\} . \end{aligned}$$

Set

$$ [\varphi ]_{s,p(x,\cdot )}=\inf \biggl\{ \mu >0: \int _{\Omega \times \Omega} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p(x,y)}}}{\mu ^{p(x,y)} \vert x-y \vert ^{N+sp(x,y)}}\,dx\,dy< 1 \biggr\} $$

as the variable exponent Gagliardo seminorm. W is a Banach space with the norm

$$ \Vert \varphi \Vert _{W}= \Vert \varphi \Vert _{W{s,q(x),p(x,\cdot )}} = \Vert \varphi \Vert _{q(x)}+[ \varphi ]_{s,p(x,\cdot )}. $$

We take into account three continuous functions $p(x,y):\overline{\Omega}\times \overline{\Omega}\rightarrow (1, \infty )$ and $r_{1}(x), r_{2}(x)\in C_{+}({\overline{\Omega}})$. From condition (P), we know that

$$ \begin{aligned} &p(x,y)=p(y,x), \quad \text{for all } x, y \in \overline{\Omega}; \\ &1< p^{-}:=\min p(x,y)\leq p(x,y)\leq p^{+}:=\max p(x,y)< \infty ; \\ &1< r_{1}^{-}:=\min r_{1}(x) \leq r_{1}(x)\leq r_{1}^{+}:=\max r_{1}(x)< \infty ; \\ &1< r_{2}^{-}:=\min r_{2}(x) \leq r_{2}(x)\leq r_{2}^{+}:=\max r_{2}(x)< \infty . \end{aligned} $$

(8)

Lemma 3.1

([34]) Suppose that $\Omega \subset \mathbb{R}^{N}$ is a bounded open domain and (8) holds. Then, W is a separable and reflexive space.

Lemma 3.2

([35]) Let smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, $sp(x, y)< N$ for $(x, y) \in \overline{\Omega}\times \overline{\Omega}$ with $s\in (0,1)$, and $q(x)\geq p(x, x)$ for $x \in \overline{\Omega}$. Suppose that continuous function $\widehat{h}(x):\overline{\Omega}\rightarrow (1,\infty )$ satisfies

$$ p^{*}_{s}(x) \geq \widehat{h}(x)\geq \widehat{h}^{-}= \inf_{x\in \Omega}\widehat{h}(x)>1, \quad \textit{for all } x\in \overline{ \Omega}. $$

There exists a positive constant $C_{0}=C_{0}(N,s,p,\widehat{h},\Omega )$ such that for every $\varphi \in W$, it holds that

$$ \Vert \varphi \Vert _{\widehat{h}(x)}\leq C_{0} \Vert \varphi \Vert _{W}. $$

Then, the embedding $W \hookrightarrow L^{\widehat{h}(x)}$ for all $\widehat{h}\in (1, p^{*}_{s})$ is compact.

Lemma 3.3

([35, 36]) If $1< sp^{-}$ and

$$ p^{*}_{\partial}(x):=\frac{(N-1){p}(x,x)}{N-s{p}(x,x)}> \widehat{h}(x), \quad \textit{in } \partial \Omega \cap \bigl\{ x\in \overline{\Omega}:N-s{p}(x,x)>0\bigr\} . $$

There exists a positive constant $C_{1}=C_{1}(N,s,p,\widehat{h},\partial \Omega )$ such that

$$ \Vert \varphi \Vert _{L^{\widehat{h}(x)}(\partial \Omega )}\leq C_{1} \Vert \varphi \Vert _{s,\overline{p}(x),p(x,y)},\quad \textit{for all } \varphi \in W^{s, \overline{p}(x),p(x,y)}(\Omega ). $$

Then, the embedding $W^{s,\overline{p}(x),p(x,y)}(\Omega )\hookrightarrow L^{\widehat{h}(x)}( \partial \Omega )$ is compact.

Define nonlinear map $\mathcal{L}:W\rightarrow W^{*}$

$$ \bigl\langle \mathcal{L}(\varphi ),\psi \bigr\rangle = \int _{\Omega} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p(x,y)-2}}(\varphi (x)-\varphi (y))(\psi (x)-\psi (y))}{ \vert x-y \vert ^{N+sp(x,y)}}\,dy, $$

(9)

for all $\varphi , \psi \in W$, $\mathcal{L}$ has the following properties.

Lemma 3.4

([28])

• (i) $\mathcal{L}$ is a bounded and strictly monotone operator;

• (ii) $\mathcal{L}$ is a mapping of $(S_{+})$, i.e., if $\varphi _{n}\rightharpoonup \varphi$ in W and $\lim_{n\rightarrow \infty}\sup \langle \mathcal{L}(\varphi _{n})- \mathcal{L}(\varphi ), \varphi _{n}-\varphi \rangle \leq 0$, then $\varphi _{n}\rightarrow \varphi$ in W;

• (iii) $\mathcal{L}:W\rightarrow W^{*}$ is a homeomorphism.

Define function $S:W\rightarrow \mathbb{R}$

$$ S(\varphi )= \int _{\Omega \times \Omega} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p(x,y)}}}{p(x,y) \vert x-y \vert ^{N+sp(x,y)}}\,dx\,dy,\quad \text{for all } \varphi \in W, $$

which is related to (9). The derivative of S is

$$\begin{aligned} \bigl\langle S'(\varphi ),\psi \bigr\rangle &= \int _{\Omega} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p(x,y)-2}}(\varphi (x)-\varphi (y))(\psi (x)-\psi (y))}{ \vert x-y \vert ^{N+sp(x,y)}}\,dy = \bigl\langle \mathcal{L}( \varphi ),\psi \bigr\rangle , \end{aligned}$$

for all $\varphi ,\psi \in W$; for more details, see [34].

Let

$$ X_{i}= \biggl\{ \varphi \in W: \int _{\Omega} \frac{V_{i}(x) \vert \varphi \vert ^{\overline{p}_{i}(x)}}{\mu ^{\overline{p}_{i}(x)}}\,dx< + \infty , \text{for some } \mu >0 \biggr\} , $$

equipped with the norm

$$ \Vert \varphi \Vert _{X_{i}}:=[\varphi ]_{s,p_{i}(x,\cdot ),\mathbb{R}^{2N} \backslash (\Omega ^{c})^{2}} +[\varphi ]_{V_{i}(\Omega )} + \bigl\Vert \beta ^{ \frac{1}{\overline{p}_{i}(x)}}\varphi \bigr\Vert _{L^{\overline{p}_{i}(x)}(C \Omega )} + \bigl\Vert \vert g \vert ^{\frac{1}{\overline{p}_{i}(x)}}\varphi \bigr\Vert _{L^{ \overline{p}_{i}(x)}(C\Omega )}, $$

where

$$ [\varphi ]_{V_{i}(\Omega )}=\inf \biggl\{ \mu >0: \int _{\Omega}V_{i}(x) \frac{ \vert \varphi (x) \vert ^{\overline{p}_{i}(x)}}{\mu ^{\overline{p}_{i}(x)}}\,dx< 1 \biggr\} , $$

and

$$ [\varphi ]_{s,p_{i}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}} =\inf \biggl\{ \mu \geq 0: \frac{1}{2} \int _{\mathbb{R}^{2N} \backslash (\Omega ^{c})^{2}} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{i}(x,y)}}}{\mu ^{p_{i}(x,y)} \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \leq 1 \biggr\} $$

with $\Omega ^{c}=\mathbb{R}^{N}\backslash \Omega$.

Lemma 3.5

([28]) Assume that assumptions (P), (G), and (V) hold. Then, $(X_{i},\|\cdot \|_{X_{i}})$ is a reflexive Banach space.

The norm $\|\cdot \|_{X_{i}}$ on $X_{i}$ is equivalent to

$$\begin{aligned} \Vert \varphi \Vert _{s,p_{i}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}} ={}&\inf \biggl\{ \mu \geq 0 | \rho _{s,p_{i}(x,\cdot ),\mathbb{R}^{2N} \backslash (\Omega ^{c})^{2}} \biggl(\frac{\varphi}{\mu}\leq 1 \biggr) \biggr\} \\ ={}&\inf \biggl\{ \mu \geq 0 | \int _{\mathbb{R}^{2N}\backslash ( \Omega ^{c})^{2}} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{p_{i}(x,y)}}{\mu ^{p_{i}(x,y)}p_{i}(x,y) \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \\ &{} + \int _{\Omega} \frac{V_{i}(x)}{\mu ^{\overline{p}_{i}(x)}\overline{p}_{i}(x)} \vert \varphi \vert ^{\overline{p}_{i}(x)}\,dx+ \int _{C\Omega} \frac{\beta (x)}{\mu ^{\overline{p}_{i}(x)}\overline{p}_{i}(x)} \vert \varphi \vert ^{\overline{p}_{i}(x)}\,dx \\ &{}+ \int _{C\Omega} \frac{g(x)}{\mu ^{\overline{p}_{i}(x)}\overline{p}_{i}(x)} \vert \varphi \vert ^{ \overline{p}_{i}x)}\,dx \leq 1 \biggr\} , \end{aligned}$$

where the modular $\rho _{s,p_{i}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}:X_{i} \rightarrow \mathbb{R}$ is defined by

$$ \begin{aligned} \rho _{s,p_{i}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}} ={}& \int _{\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{i}(x,y)}}}{p_{i}(x,y) \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy + \int _{\Omega}\frac{V_{i}(x)}{\overline{p}_{i}(x)} \vert \varphi \vert ^{ \overline{p}_{i}(x)}\,dx \\ &{}+ \int _{C\Omega}\frac{\beta (x)}{\overline{p}_{i}(x)} \vert \varphi \vert ^{ \overline{p}_{i}(x)}\,dx + \int _{C\Omega} \frac{g(x)}{\overline{p}_{i}(x)} \vert \varphi \vert ^{\overline{p}_{i}(x)}\,dx. \end{aligned} $$

(10)

Lemma 3.6

Assume that assumptions (P), (G), and (V) hold. The following properties hold:

• (i) $\|\varphi \|_{X_{i}}<1(=1, >1) \Longleftrightarrow \rho _{s,p_{i}(x, \cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}(\varphi )<1 (=1, >1)$;

• (ii) ${\parallel \phi \parallel }_{X_i}\ge 1⟹{\parallel \phi \parallel }_{X_i}^{p_i^{-}}\le {\rho }_{s,p_i(x,\cdot ),{\mathbb{R}}^{2N}\mathrm{\setminus }{({\mathrm{\Omega }}^c)}^2}(\phi )\le {\parallel \phi \parallel }_{X_i}^{p_i^{+}}$;

• (iii) ${\parallel \phi \parallel }_{X_i}\le 1⟹{\parallel \phi \parallel }_{X_i}^{p_i^{+}}\le {\rho }_{s,p_i(x,\cdot ),{\mathbb{R}}^{2N}\mathrm{\setminus }{({\mathrm{\Omega }}^c)}^2}(\phi )\le {\parallel \phi \parallel }_{X_i}^{p_i^{-}}$;

• (iv) $\rho _{s,p_{i}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}( \varphi -v)\rightarrow 0 \Leftrightarrow \|\varphi -v\|_{X_{i}} \rightarrow 0$.

Let $X=X_{1}\cap X_{2}$ with norm $\|\varphi \|_{X}=\|\varphi \|_{X_{1}}+\|\varphi \|_{X_{2}}$, which is a separable and reflexive Banach space. The dual space of X is $X^{*}$. The modular $\rho _{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}= \rho _{s,p_{1}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}+ \rho _{s,p_{2}(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}$. We have the following result.

Lemma 3.7

([28]) Assume that assumptions (P), (G), and (V) hold. Then, from (10), the following properties hold:

• (i) The function $\rho _{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2})}$ is of class $C^{1}(X,\mathbb{R})$;

• (ii) The strictly monotone operator $\rho '_{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}: X \rightarrow X^{*}$ is coercive, then

  $$ \frac{\langle \rho '_{s,p(x,\cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}},\varphi \rangle _{X}}{ \Vert \varphi \Vert _{X}}\rightarrow +\infty , \qquad \Vert \varphi \Vert _{X} \rightarrow +\infty ; $$

• (iii) $\rho '_{s,p(x, \cdot ),\mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}}$ is a mapping of type $(S_{+})$, that is, if $\varphi _{n}\rightharpoonup \varphi$ in X and $\lim \sup_{n\rightarrow +\infty} \langle \rho '_{s,p(x,\cdot ), \mathbb{R}^{2N}\backslash (\Omega ^{c})^{2}},\varphi \rangle _{X} \leq 0$, then $\varphi _{n}\rightarrow \varphi$ in X.

Lemma 3.8

([35, 36]) Assume that assumptions (P), (G), (V), and (H) hold. Then, for any $\widehat{r}\in C_{+}(\overline{\Omega})$ with $1<\widehat{r}(x)<p_{s}^{*}(x)$ for all $x\in \Omega$, there is a positive constant $\varpi ^{*}=\varpi ^{*}(s,p_{i},N,\widehat{r},\Omega )>0$ such that

$$ \Vert \varphi \Vert _{L^{\widehat{r}(x)}(\Omega )}\leq \varpi ^{*} \Vert \varphi \Vert _{X},\quad \textit{for all } \varphi \in X. $$

Moreover, this embedding is compact.

Lemma 3.9

([35]) Assume that assumptions (P), (G), (V), and (H) hold. Then, for any $\widehat{r}\in C_{+}(\mathbb{R}^{N}\backslash \Omega )$ with $1<\widehat{r}(x)<p_{\partial}^{*}(x)$ for all $x\in \mathbb{R}^{N}\backslash \Omega$, there is a positive constant $\widehat{\varpi}^{*}=\widehat{\varpi}^{*}(s,p_{i},N,\widehat{r}, \partial \Omega )>0$ such that

$$ \Vert \varphi \Vert _{L^{\widehat{r}(x)}(\mathbb{R}^{N}\backslash \Omega )} \leq \widehat{\varpi}^{*} \Vert \varphi \Vert _{X},\quad \textit{for all } \varphi \in X. $$

Moreover, this embedding is compact.

More precisely, we now present the divergence theorem and the analogous formula for the partition integral formula in nonlocal case [37].

Lemma 3.10

([29]) Let the hypotheses (P) hold, and let φ be any bounded $C^{2}$-function in $R^{N}$. Then,

$$\begin{aligned} & \int _{\Omega} (-\Delta )_{p_{1}(x,\cdot )}^{s} \varphi (x)\,dx + \int _{\Omega} (-\Delta )_{p_{2}(x, \cdot )}^{s}\varphi (x)\,dx \\ &\quad =- \biggl( \int _{\mathbb{R}^{N}\setminus \Omega}\mathcal{N}_{s,p_{1}(x, \cdot )}\varphi (x)\,dx + \int _{\mathbb{R}^{N}\setminus \Omega} \mathcal{N}_{s,p_{2}(x,\cdot )}\varphi (x)\,dx \biggr). \end{aligned}$$

Lemma 3.11

Let the hypotheses (P) hold. Suppose that φ and v are bounded $C^{2}$-functions in $\mathbb{R}^{N}$. Then,

$$\begin{aligned} &\frac{1}{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (C \Omega )^{2}} \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{p_{1}(x,y)-2} \frac{(\varphi (x)-\varphi (y))(v(x)-v(y))}{ \vert x-y \vert ^{N+sp_{1}(x,y)}}\,dx\,dy \\ & \qquad {}+ \int _{\mathbb{R}^{2N}\setminus (C\Omega )^{2}} \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{p_{2}(x,y)-2} \frac{(\varphi (x)-\varphi (y))(v(x)-v(y))}{ \vert x-y \vert ^{N+sp_{2}(x,y)}}\,dx\,dy \biggr) \\ &\quad = \int _{\Omega}v (-\Delta )_{p_{1}(x,\cdot )}^{s}\varphi (x)\,dx+ \int _{ \Omega}v(-\Delta )_{p_{2}(x,\cdot )}^{s}\varphi (x)\,dx \\ &\qquad {}+ \int _{C\Omega}v\mathcal{N}_{s,p_{1}(x,\cdot )}\varphi (x)\,dx+ \int _{C \Omega}v\mathcal{N}_{s,p_{2}(x,\cdot )}\varphi (x)\,dx, \end{aligned}$$

for every $v\in X$.

Proof

According to symmetry, we obtain

$$\begin{aligned} &\frac{1}{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (C \Omega )^{2}} \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{p_{1}(x,y)-2} \frac{(\varphi (x)-\varphi (y))(v(x)-v(y))}{ \vert x-y \vert ^{N+sp_{1}(x,y)}}\,dx\,dy \\ &\qquad {} + \int _{\mathbb{R}^{2N}\setminus (C\Omega )^{2}} \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{p_{2}(x,y)-2} \frac{(\varphi (x)-\varphi (y))(v(x)-v(y))}{ \vert x-y \vert ^{N+sp_{2}(x,y)}}\,dx\,dy \biggr) \\ &\quad = \int _{\Omega} \int _{\mathbb{R}^{N}} v(x) \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{{p_{1}(x,y)-2}} \frac{\varphi (x)-\varphi (y)}{ \vert x-y \vert ^{N+sp_{1}(x,y)}}\,dy\,dx \\ &\qquad {}+ \int _{\Omega} \int _{\mathbb{R}^{N}} v(x) \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{{p_{2}(x,y)-2}} \frac{\varphi (x)-\varphi (y)}{ \vert x-y \vert ^{N+sp_{2}(x,y)}}\,dy\,dx \\ &\qquad {}+ \int _{C\Omega} \int _{\Omega} v(x) \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{{p_{1}(x,y)-2}} \frac{\varphi (x)-\varphi (y)}{ \vert x-y \vert ^{N+sp_{1}(x,y)}}\,dy\,dx \\ &\qquad {}+ \int _{C\Omega} \int _{\Omega} v(x) \bigl\vert \varphi (x)-\varphi (y) \bigr\vert ^{{p_{2}(x,y)-2}} \frac{\varphi (x)-\varphi (y)}{ \vert x-y \vert ^{N+sp_{2}(x,y)}}\,dy\,dx. \end{aligned}$$

 □

Lemma 3.12

Assuming that assumption (P) holds and letting φ be a weak solutions of equations (1), we have

$$ \begin{aligned}&\mathcal{N}_{s,p_{1}(x,\cdot )}\varphi +\mathcal{N}_{s,p_{2}(x,\cdot )} \varphi +\beta (x) \vert \varphi \vert ^{\bar{p}_{1}(x)-2}\varphi +\beta (x) \vert \varphi \vert ^{\bar{p}_{2}(x)-2}\varphi \\ &\quad =g_{1}(x)+g_{2}(x), \quad \textit{a.e. in } \mathbb{R}^{N}\setminus \overline{\Omega}. \end{aligned}$$

Lemma 3.13

Assuming that assumptions (P), (G), (V), and (H) hold, let $I_{\lambda}: X \rightarrow \mathbb{R}$ be a energy function defined by

$$\begin{aligned} I_{\lambda}(\varphi )={}& \frac{1}{2} \int _{\mathbb{R}^{2N}\setminus (C \Omega )^{2}}\frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{1}(x,y)}}}{p_{1}(x,y) \vert x-y \vert ^{N+sp_{1}(x,y)}}\,dx\,dy \\ &{}+\frac{1}{2} \int _{\mathbb{R}^{2N}\setminus (C\Omega )^{2}} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{2}(x,y)}}}{p_{2}(x,y) \vert x-y \vert ^{N+sp_{2}(x,y)}}\,dx\,dy \\ &{}+ \int _{\Omega}V_{1}(x) \frac{ \vert \varphi \vert ^{\bar{p}_{1}(x)}}{\bar{p}_{1}(x)}\,dx + \int _{\Omega}V_{2}(x) \frac{ \vert \varphi \vert ^{\bar{p}_{2}(x)}}{\bar{p}_{2}(x)}\,dx + \int _{ \mathbb{R}^{N}\setminus \Omega}\beta (x) \frac{ \vert \varphi \vert ^{\bar{p}_{1}(x)}}{\bar{p}_{1}(x)}\,dx \\ &{}+ \int _{\mathbb{R}^{N}\setminus \Omega}\beta (x) \frac{ \vert \varphi \vert ^{\bar{p}_{2}(x)}}{\bar{p}_{2}(x)}\,dx -\lambda _{1} \int _{\Omega}A_{1}(x)\frac{ \vert \varphi \vert ^{r_{1}(x)}}{r_{1}(x)}\,dx - \lambda _{2} \int _{\Omega}A_{2}(x) \frac{ \vert \varphi \vert ^{r_{2}(x)}}{r_{2}(x)}\,dx \\ &{}- \int _{\mathbb{R}^{N}\setminus \Omega}g_{1}(x)\varphi \,dx - \int _{ \mathbb{R}^{N}\setminus \Omega}g_{2}(x)\varphi \,dx, \end{aligned}$$

for every $\varphi \in X$. Then, any critical point of $I_{\lambda}$ is a weak solution of equations (1).

4 Proof of Theorem 1.1

To prove Theorem 1.1, we need a well-known mountain pass lemma.

Theorem 4.1

Let X be a real Banach space and $I_{\lambda}\in C^{1}(X, \mathbb{R})$ with $I_{\lambda}(0)=0$. Assume that the following conditions hold:

• (i) $I_{\lambda}$ satisfies $(PS )$ conditions;

• (ii) there exist $\rho , \sigma >0$ such that $I_{\lambda}(\varphi )\geq \sigma$, for all $\varphi \in X$, with $\|\varphi \|_{X}=\rho$;

• (iii) there exists $\upsilon \in X$, satisfying $\|\upsilon \|_{X}>\rho$ such that $I_{\lambda}(\upsilon )<0$.

Then, $I_{\lambda}$ has a critical value $c>\sigma$, that is,

$$ c=\inf_{\gamma \in \Upsilon}\max_{0\leq t\leq 1}I_{\lambda} \bigl(\gamma (t)\bigr), $$

where $\Upsilon =\{\gamma \in C^{1}([0,1];X):\gamma (0)=1,\gamma (1)= \upsilon \}$.

Definition 1

Let X be a Banach space, $I_{\lambda}\in C^{1}(E,\mathbb{R})$. We say that $I_{\lambda}$ satisfies the $(PS )$ conditions if every sequence $\{\varphi _{n}\}_{n\in \mathbb{N}}\subset X$ satisfying

$$ I_{\lambda}(\varphi _{n})\rightarrow c, \qquad I_{\lambda}( \varphi _{n}) \rightarrow 0,\quad n\rightarrow \infty $$

has a convergent subsequence in X.

Next, we prove that the $I_{\lambda}$ defined in Lemma 3.13 satisfies the $(PS)$ conditions.

Lemma 4.1

Assume that assumptions (P), (G), (V), and (H) hold. Then, the sequence $\{\varphi _{n}\}_{n\in \mathbb{N}}$ is bounded in X.

Proof

According to (H), we get

$$ s'_{1}(x)r_{1}(x)< p^{*}_{s}(x),\qquad s'_{2}(x)r_{2}(x)< p^{*}_{s}(x), \quad \text{for all } x\in \Omega , $$

so from Lemmas 3.8 and 3.9, there exist constants $M_{1}$ and $M_{2}$ such that

$$ \Vert \varphi \Vert _{L^{s'_{1}(x)r_{1}(x)}(\Omega )}\leq M_{1} \Vert \varphi \Vert _{X},\qquad \Vert \varphi \Vert _{L^{s'_{2}(x)r_{2}(x)}(\Omega )} \leq M_{2} \Vert \varphi \Vert _{X}, \quad \text{for all } \varphi \in X. $$

(11)

Let $\rho >\max \{1,\frac{1}{M_{1}},\frac{1}{M_{2}} \}$ and

$$ \Vert \varphi \Vert _{L^{s'_{1}(x)r_{1}(x)}(\Omega )}> 1,\qquad \Vert \varphi \Vert _{L^{s'_{2}(x)r_{2}(x)}( \Omega )}> 1. $$

Thus, by the Hölder inequality and Lemma 2.3, for all $\varphi \in X$ with $\|\varphi \|_{X}=\rho$, we obtain

$$ \begin{aligned} \int _{\Omega}A_{1}(x) \bigl\vert \varphi (x) \bigr\vert ^{r_{1}(x)}\,dx &\leq 2 \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert ^{r_{1}^{+}}_{s'_{1}(x)r_{1}(x)} \leq 2M_{1}^{r_{1}^{+}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert ^{r_{1}^{+}}_{X} \end{aligned} $$

(12)

and

$$ \begin{aligned} \int _{\Omega}A_{2}(x) \bigl\vert \varphi (x) \bigr\vert ^{r_{2}(x)}\,dx &\leq 2 \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert ^{r_{2}^{+}}_{s'_{2}(x)r_{2}(x)} \leq 2M_{2}^{r_{2}^{+}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert ^{r_{2}^{+}}_{X}. \end{aligned} $$

(13)

We use the counterfactual method. Suppose $\|\varphi _{n}\|_{X}\rightarrow \infty$, $n\rightarrow \infty$. Combining conditions (P), (G), (V), (H), and Lemma 3.8 and letting $0<\vartheta <\min \{r_{1}^{-}r_{2}^{-} \frac{Z_{A} +Z_{B}-1}{r_{2}^{-}Z_{A}K_{A}+ r_{1}^{-}Z_{B}K_{A}}, r_{1}^{-},r_{2}^{-} \}$, where $K_{A}=\max \{\|A_{1}\|_{X}, \|A_{2}\|_{X}\}$, $Z_{A}=4\lambda _{1}M_{1}^{r_{1}^{+}}C_{A_{1}}$, $Z_{B}=4\lambda _{2}M_{2}^{r_{2}^{+}}C_{A_{2}}$, we have

$$\begin{aligned} &c\frac{1}{ \Vert \varphi _{n} \Vert ^{p_{1}^{-}}_{X}}+o(1) \frac{1}{ \Vert \varphi _{n} \Vert ^{p_{1}^{-}}_{X}} \\ &\quad \geq \frac{1}{ \Vert \varphi _{n} \Vert ^{p_{1}^{-}}_{X}} \biggl(I_{\lambda}( \varphi _{n})-\biggl\langle \frac{1}{\vartheta}I'_{\lambda}(\varphi _{n}), \varphi _{n} \biggr\rangle \biggr) \\ &\quad =\frac{1}{ \Vert \varphi _{n} \Vert ^{p_{1}^{-}}_{X}} \sum_{i=1} ^{2} \biggl[ \int _{\mathbb{R}^{2N}\setminus (\mathcal{C} \Omega )^{2}} \frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{i}(x,y)}}}{2p_{i}(x,y) \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy + \int _{\Omega}V_{i}(x) \frac{ \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \\ &\qquad {} + \int _{\mathbb{R}^{N}\setminus \Omega}\beta (x) \frac{ \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx -\lambda _{1} \int _{\Omega}A_{1}(x)\frac{ \vert \varphi _{n} \vert ^{r_{1}(x)}}{r_{1}(x)}\,dx - \lambda _{2} \int _{\Omega}A_{2}(x) \frac{ \vert \varphi _{n} \vert ^{r_{2}(x)}}{r_{2}(x)}\,dx \\ &\qquad {}- \int _{\mathbb{R}^{N}\setminus \Omega}g_{i}(x)\varphi _{n}\,dx - \frac{1}{\vartheta} \biggl( \int _{\mathbb{R}^{2N}\setminus ( \mathcal{C}\Omega )^{2}}\frac{ \vert \varphi _{n}(x)-\varphi _{n}(y) \vert ^{p_{i}(x,y)}}{2 \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \\ &\qquad {} + \int _{\Omega}V_{i}(x) \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)}\,dx + \int _{\mathbb{R}^{N}\setminus \Omega}\beta (x) \vert \varphi _{n} \vert ^{ \bar{p}_{i}(x)}\,dx -\lambda _{1} \int _{\Omega}A_{1}(x) \vert \varphi _{n} \vert ^{r_{1}(x)}\,dx \\ & \qquad {} -\lambda _{2} \int _{\Omega}A_{2}(x) \vert \varphi _{n} \vert ^{r_{2}(x)}\,dx - \int _{\mathbb{R}^{N}\setminus \Omega}g_{i}(x)\varphi _{n}\,dx \biggr) \biggr] \\ &\quad \geq \frac{1}{ \Vert \varphi _{n} \Vert ^{p_{1}^{-}}_{X}} \Biggl[ \biggl( \frac{1}{2p_{2}^{+}}-\frac{1}{2\vartheta} \biggr) \sum_{i=1} ^{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (\mathcal{C} \Omega )^{2}} \frac{ \vert \varphi _{n}(x)-\varphi _{n}(y) \vert ^{p_{i}(x,y)}}{ \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \\ & \qquad {} + \int _{\Omega}V_{i}(x) \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)}\,dx + \int _{\mathbb{R}^{N}\setminus \Omega}\beta (x) \vert \varphi _{n} \vert ^{ \bar{p}_{i}(x)}\,dx - \int _{\mathbb{R}^{N}\setminus \Omega}g_{i}(x) \varphi _{n}\,dx \biggr) \\ &\qquad {}- 2\bigl(\vartheta -r_{1}^{+}\bigr) \frac{\lambda _{1}M_{1}^{r_{1}^{+}}}{\vartheta r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi _{n} \Vert ^{r_{1}^{+}}_{X}- 2\bigl(\vartheta -r_{2}^{+}\bigr) \frac{\lambda _{2}M_{2}^{r_{2}^{+}}}{\vartheta r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi _{n} \Vert ^{r_{2}^{+}}_{X} \Biggr] \\ &\quad \geq \biggl(\frac{1}{2p_{2}^{+}}-\frac{1}{2\vartheta} \biggr) -2\bigl( \vartheta -r_{1}^{-}\bigr) \frac{\lambda _{1}M_{1}^{r_{1}^{+}}C_{A_{1}}}{\vartheta r_{1}^{-}} \Vert A_{1} \Vert _{X} -2\bigl(\vartheta -r_{2}^{-}\bigr) \frac{\lambda _{2}M_{2}^{r_{2}^{+}}C_{A_{2}}}{\vartheta r_{2}^{-}} \Vert A_{2} \Vert _{X}. \end{aligned}$$

In addition, we obtain $c\frac{1}{\|\varphi _{n}\|^{p_{1}^{-}}_{X}}+o(1) \frac{1}{\|\varphi _{n}\|^{p_{1}^{-}}_{X}}\rightarrow 0$ because $\|\varphi _{n}\|_{X}\rightarrow \infty$, $n\rightarrow \infty$. Due to

$$ 0< \vartheta < \min \biggl\{ r_{1}^{-}r_{2}^{-} \frac{Z_{A} +Z_{B}-1}{ r_{2}^{-}Z_{A}K_{A}+ r_{1}^{-}Z_{B}K_{A}}, r_{1}^{-}, r_{2}^{-} \biggr\} , $$

there is a contradiction. Thus, $\{\varphi _{n}\}_{n\in \mathbb{N}}$ is bounded. □

Inspired by [15], we have the following lemma.

Lemma 4.2

Assume that assumptions (P), (G), (V), and (H) hold. Then, $I_{\lambda}$ satisfies the $(PS)$ conditions.

Proof

According to Lemma 4.1, $\{\varphi _{n}\}_{n\in \mathbb{N}}$ is bounded, that is, there is a subsequence $\{\varphi _{n}\}_{n\in \mathbb{N}}$ and $\varphi _{0}$ in X such that

$$\begin{aligned} &\varphi _{n}\rightharpoonup \varphi _{0} \quad \text{in } X; \\ &\varphi _{n}\rightarrow \varphi _{0} \quad \text{a.e. in } \Omega ; \\ &\varphi _{n}\rightarrow \varphi _{0} \quad \text{in } L^{\widehat{r}(x)}( \Omega ), \widehat{r}(x)< p^{*}_{s}(x); \\ &\varphi _{n}\rightarrow \varphi _{0} \quad \text{in } L^{\widehat{r}(x)}\bigl( \mathbb{R}^{N}\setminus \Omega \bigr), \widehat{r}(x)< p^{*}_{\partial}(x). \end{aligned}$$

Due to $\varphi _{n}\rightarrow \varphi _{0}$ in $L^{\widehat{r}(x)}(\mathbb{R}^{N}\setminus \Omega )$, then $|\varphi _{n}|^{\bar{p}_{i}(x)-2}\varphi _{n}\rightarrow |\varphi _{0}|^{ \bar{p}_{i}(x)-2}\varphi _{0}$. We get

$$ \lim_{n\rightarrow \infty} \int _{\mathbb{R}\setminus \Omega}\beta (x) \bigl( \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)-2}\varphi _{n} - \vert \varphi _{0} \vert ^{ \bar{p}_{i}(x)-2}\varphi _{0} \bigr)=0. $$

Since $\lambda _{1} A_{1}(x)|\varphi |^{r_{1}(x)-2}\varphi$ and $\lambda _{2}A_{2}(x)|\varphi |^{r_{2}(x)-2}\varphi$ in X are sequentially weakly lower semi-continuous, for $v\in X$ and measurable for all $\Omega \subset \mathbb{R}^{N}$, we obtain

$$\begin{aligned} & \biggl\vert \int _{\Omega}A_{1}(x) \bigl( \vert \varphi \vert _{n}^{r_{1}(x)-2}\varphi _{n}- \vert \varphi \vert _{0}^{r_{1}(x)-2}\varphi _{0}\bigr) \biggr\vert v \vert \,dx \vert \\ &\quad \leq \int _{ \Omega}A_{1}(x) \bigl( \vert \varphi \vert _{n}^{r_{1}(x)-1}- \vert \varphi \vert _{0}^{r_{1}(x)-1} \bigr)v\,dx \\ &\quad = \int _{\Omega}A_{1}(x)^{\frac{r_{1}(x)-1}{r_{1}(x)}}\bigl( \vert \varphi \vert _{n}^{r_{1}(x)-1}- \vert \varphi \vert _{0}^{r_{1}(x)-1}\bigr) A_{1}(x)^{\frac{1}{r_{1}(x)}} \vert v \vert \,dx \\ &\quad \leq \big\Vert A_{1}(x)^{\frac{r_{1}(x)-1}{r_{1}(x)}}\bigl( \vert \varphi \vert _{n}^{r_{1}(x)-1}- \vert \varphi \vert _{0}^{r_{1}(x)-1} \big\| _{L^{\frac{r_{1}(x)}{r_{1}(x)-1}}} \bigl\Vert A_{1}(x)^{ \frac{1}{r_{1}(x)}}\bigr) \vert v \vert \bigr\Vert _{L^{r_{1}(x)}}. \end{aligned}$$

Hence, $\{A_{1}(x)(|\varphi |_{n}^{r_{1}(x)-2}\varphi _{n}- |\varphi |_{0}^{r_{1}(x)-2} \varphi _{0})|v|\}_{n\in \mathbb{N}}$ is uniformly integrable in $\mathbb{R}^{N}$. Then, using the Vitali convergence theorem, we get

$$ \lim_{n\rightarrow \infty} \int _{\Omega}A_{1}(x) \bigl( \vert \varphi _{n} \vert ^{r_{1}(x)-2} \varphi _{n} - \vert \varphi _{0} \vert ^{r_{1}(x)-2}\varphi _{0} \bigr)=0. $$

Similarly, there is

$$ \lim_{n\rightarrow \infty} \int _{\Omega}A_{2}(x) \bigl( \vert \varphi _{n} \vert ^{r_{2}(x)-2} \varphi _{n} - \vert \varphi _{0} \vert ^{r_{2}(x)-2}\varphi _{0} \bigr)=0. $$

We need to prove that $\{\varphi _{n}\}_{n\in \mathbb{N}}$ is strongly convergent,

$$\begin{aligned} o(1)={}&\bigl\langle I'_{\lambda}(\varphi _{n})-I'_{\lambda}( \varphi _{0}), \varphi _{n}-\varphi _{0} \bigr\rangle \\ ={}& \sum_{i=1} ^{2} \biggl[ \int _{\mathbb{R}^{2N} \setminus (\mathcal{C}\Omega )^{2}} \frac{ \vert \varphi _{n}(x)-\varphi _{n}(y) \vert ^{p_{i}(x,y)-2}(\varphi _{n}(x)-\varphi _{n}(y)) (\varphi _{n}(x)-\varphi _{n}(y)-\varphi _{0}(x)+\varphi _{0}(y))}{ 2 \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \\ &{}- \int _{\mathbb{R}^{2N}\setminus (\mathcal{C}\Omega )^{2}} \frac{ \vert \varphi _{0}(x)-\varphi _{0}(y) \vert ^{p_{i}(x,y)-2}(\varphi _{0}(x)-\varphi _{0}(y)) (\varphi _{n}(x)-\varphi _{n}(y)-\varphi _{0}(x)+\varphi _{0}(y))}{ 2 \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \\ &{}+ \int _{\Omega}V_{i}(x) \bigl( \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)-2}\varphi _{n}- \vert \varphi _{0} \vert ^{\bar{p}_{i}(x)-2}\varphi _{0}\bigr) (\varphi _{n}-\varphi _{0})\,dx \\ &{}+ \int _{\mathbb{R}^{N}\setminus \Omega}\beta (x) \bigl( \vert \varphi _{n} \vert ^{ \bar{p}_{i}(x)-2}\varphi _{n}- \vert \varphi _{0} \vert ^{\bar{p}_{i}(x)-2} \varphi _{0}\bigr) (\varphi _{n}- \varphi _{0})\,dx \\ &{}-\lambda _{1} \int _{\Omega}A_{1}(x) \bigl( \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)-2} \varphi _{n}- \vert \varphi _{0} \vert ^{\bar{p}_{i}(x)-2}\varphi _{0}\bigr) (\varphi _{n}- \varphi _{0})\,dx \\ &{} -\lambda _{2} \int _{\Omega}A_{2}(x) \bigl( \vert \varphi _{n} \vert ^{\bar{p}_{i}(x)-2} \varphi _{n}- \vert \varphi _{0} \vert ^{\bar{p}_{i}(x)-2}\varphi _{0}\bigr) (\varphi _{n}- \varphi _{0})\,dx \biggr] \\ ={}&\sum_{i=1} ^{2} \biggl[ \int _{\mathbb{R}^{2N} \setminus (\mathcal{C}\Omega )^{2}} \frac{ \vert \varphi _{n}(x)-\varphi _{n}(y) \vert ^{p_{i}(x,y)-2}(\varphi _{n}(x)-\varphi _{n}(y)) (\varphi _{n}(x)-\varphi _{n}(y)-\varphi _{0}(x)+\varphi _{0}(y))}{ 2 \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \\ &{} - \int _{\mathbb{R}^{2N}\setminus (\mathcal{C}\Omega )^{2}} \frac{ \vert \varphi _{0}(x)-\varphi _{0}(y) \vert ^{p_{i}(x,y)-2}(\varphi _{0}(x)-\varphi _{0}(y)) (\varphi _{n}(x)-\varphi _{n}(y)-\varphi _{0}(x)+\varphi _{0}(y))}{ 2 \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy \biggr]. \end{aligned}$$

A discussion similar to Lemma 3.7 gives that $\varphi _{n}\rightarrow \varphi _{0}$ in X. Combining the Definition 1 and the Lemma 4.1, we complete the proof. □

Lemma 4.3

Assume that assumptions (P), (G), (V), and (H) hold. There exist $\rho >0$ and $\sigma >0$ such that, for all $\varphi \in X$ with $\|\varphi \| _{X}=\rho$,

$$ I_{\lambda}(\varphi )\geq \sigma >0 $$

holds.

Proof

Combining (12) with (13), for any $\varphi \in X$ with $\|\varphi \|_{X}=\rho >1$, we have

$$\begin{aligned} I_{\lambda}(\varphi )\geq {}& \frac{1}{p_{2}^{+}}\sum _{i=1} ^{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (\mathcal{C} \Omega )^{2}}\frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{i}(x,y)}}}{2 \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy+ \int _{\Omega}V_{i}(x) \vert \varphi \vert ^{\bar{p}_{i}(x)}\,dx \\ & {}+ \int _{\mathbb{R}^{N}\setminus \Omega}\beta (x) \vert \varphi \vert ^{ \bar{p}_{i}(x)}\,dx - \int _{\mathbb{R}^{N}\setminus \Omega}g_{i}(x) \varphi \,dx \biggr) \\ &{}-\frac{\lambda _{1}}{r_{1}^{-}} \int _{\Omega}A_{1}(x) \vert \varphi \vert ^{r_{1}(x)}\,dx -\frac{\lambda _{2}}{r_{2}^{-}} \int _{\Omega}A_{2}(x) \vert \varphi \vert ^{r_{2}(x)}\,dx \\ \geq {}&\frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{2}^{+}} -\lambda _{1} \frac{2M_{1}^{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)}}{r_{1}^{-}} \Vert \varphi \Vert _{X}^{r_{1}^{-}} -\lambda _{2}\frac{2M_{2}^{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)}}{r_{2}^{-}} \Vert \varphi \Vert _{X}^{r_{2}^{-}} \\ ={}& \Vert \varphi \Vert _{X}^{r_{1}^{-}} \biggl( \frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{2}^{+}-r_{1}^{-}} - \lambda _{2}\frac{2M_{2}^{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)}}{r_{2}^{-}} \Vert \varphi \Vert _{X}^{r_{2}^{-}-r_{1}^{-}} -\lambda _{1} \frac{2M_{1}^{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)}}{r_{1}^{-}} \biggr). \end{aligned}$$

Let

$$ f(t)=\frac{1}{p_{2}^{+}}t^{p_{2}^{+}-r_{1}^{-}} -\lambda _{2} \frac{2M_{2}^{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)}}{r_{2}^{-}}t^{r_{2}^{-}-r_{1}^{-}},\quad t \geq 0, $$

where $r_{1}^{-}< r_{2}^{-}< p_{2}^{+}$. Then, there exists $\chi >0$ such that $f(t)=f(\chi )>0$. Choosing $\|A_{1}\|_{s_{1}(x)}^{{r_{1}^{-}}}<\sigma ^{*}=\frac{r_{1}^{-}}{4\lambda _{1}M_{1}^{r_{1}^{-}}}f(\chi )$, we get

$$ I_{\lambda}(\varphi )\geq \sigma =\chi ^{r_{1}^{-}} \frac{f(\chi )}{2}>0, $$

for $\|\varphi \|_{X}=\chi = \rho$. □

Lemma 4.4

Assume that assumptions (P), (G), (V), and (H) hold. Then, there exists υ, which satisfies $\|\upsilon \|_{X}>\rho$. Then, there exists $\upsilon \in X$ such that

$$ I_{\lambda}(\upsilon )< 0. $$

Proof

Choosing $\widehat{\upsilon}\in X$ such that $\|\widehat{\upsilon}\|_{X}=1$, and for $t\in (0,1)$ small enough, we obtain

$$\begin{aligned} I_{\lambda}(t\widehat{\upsilon}) \leq {}& t^{p_{2}^{+}}\sum _{i=1} ^{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (\mathcal{C} \Omega )^{2}}\frac{ \vert \widehat{\upsilon}(x)-\widehat{\upsilon}(y) \vert ^{{p_{i}(x,y)}}}{2{p_{i}}(x,y) \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy + \int _{\Omega}V_{i}(x) \frac{ \vert \widehat{\upsilon} \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \\ & {}+ \int _{\mathbb{R}^{N}\setminus \Omega} \frac{\beta (x) \vert \widehat{\upsilon} \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \biggr) -\lambda _{1}t^{{r_{1}}^{-}} \int _{\Omega} \frac{A_{1}(x) \vert \widehat{\upsilon} \vert ^{r_{1}(x)}}{r_{1}(x)}\,dx \\ &{}-\lambda _{2}t^{{r_{2}}^{-}} \int _{\Omega} \frac{A_{2}(x) \vert \widehat{\upsilon} \vert ^{r_{2}(x)}}{r_{2}(x)}\,dx - \int _{ \mathbb{R}^{N}\setminus \Omega}t\widehat{\upsilon} g_{1}(x)\,dx - \int _{ \mathbb{R}^{N}\setminus \Omega}t\widehat{\upsilon} g_{2}(x)\,dx < 0 \end{aligned}$$

with the fact that $1< r_{1}^{-}< p_{2}^{+}$. Thus, $I_{\lambda}(t\widehat{\upsilon})<0$ with $\|t\widehat{\upsilon}\|_{X}>\rho$. The proof is proved by letting $\upsilon =t\widehat{\upsilon}$. □

Proof of Theorem 1.1

Combining Lemmas 4.1 and 4.2, it can be inferred that $I_{\lambda}$ satisfies $(PS)$ conditions. According to Lemmas 4.3 and 4.4, we know that $I_{\lambda}$ satisfies the mountain pass lemma. Therefore, we have a subsequence $\{\varphi _{n}\}_{n\in \mathbb{N}}$ and $\varphi _{0}^{(1)}\in X$ such that $\varphi _{n}\rightarrow \varphi _{0}^{(1)}$ in X by Lemma 4.1 and $0<\sigma <c<\infty$. Therefore, $I_{\lambda}(\varphi _{n})= c>\sigma$, that is, $\varphi _{0}^{(1)}$ is a solution of problem (1) with positive energy.

Next, we will apply Ekeland’s variational principle to prove that (1) has a solution with negative energy.

By Lemma 4.3, we derive that

$$ \widehat{c}=\inf_{\partial B_{\rho}(0)} I_{\lambda}>0, $$

where ρ is the positive constant introduced in Lemma 4.3.

From condition (H), there exist $\epsilon _{1},\epsilon _{2}>0$ and an open set $\Omega _{0}\subset \subset \Omega$ such that

$$ \bigl\vert r_{1}(x)-{r_{1}}^{-} \bigr\vert < \epsilon _{1}, \qquad \bigl\vert r_{2}(x)-{r_{2}}^{-} \bigr\vert < \epsilon _{2}, \quad \text{for all } x\in \Omega _{0}, $$

and we get

$$ {r_{1}}^{-}+\epsilon _{1}< p_{1}^{-},\qquad {r_{2}}^{-}+\epsilon _{2}< p_{1}^{-}, \quad \text{for all } x\in \overline{\Omega}_{0}. $$

Hence,

$$ r_{1}(x)\leq{r_{1}}^{-}+\epsilon _{1}< p_{1}^{-},\qquad r_{2}(x) \leq{r_{2}}^{-}+ \epsilon _{2}< p_{1}^{-}, \quad \text{for all } x\in \overline{\Omega}_{0}. $$

(14)

By Lemma 2.1 and $g_{i}(x)>0$, we conclude

$$\begin{aligned} \int _{\mathbb{R}\setminus \Omega}g_{i}\varphi \,dx \leq {}& \int _{ \mathbb{R}\setminus \Omega} \vert g_{i} \vert ^{\frac{1}{\overline{p}_{i}'(x)}} \vert g_{i} \vert ^{ \frac{1}{\overline{p}_{i}(x)}} \vert \varphi \vert \,dx \\ \leq {}& 2 \Vert g_{i} \Vert _{L^{1}(\mathbb{R}\setminus \Omega )} \bigl\Vert \vert g_{i} \vert ^{ \frac{1}{\overline{p}_{i}(x)}} \vert \varphi \vert \bigr\Vert _{L^{\overline{p}_{i}(x)}( \mathbb{R}\setminus \Omega )} \leq K_{1} \Vert \varphi \Vert _{X}. \end{aligned}$$

For sufficiently small $\tau \in (0,1)$, let $\eta \in C_{0}^{\infty}(\Omega )$ such that $\overline{\Omega}_{0}\subset \operatorname{supp}(\eta )$, $\eta =1$, for all $x\in \overline{\Omega}_{0}$ and $0\leq \eta \leq 1$ in Ω. Then, by applying (14), it follows that

$$\begin{aligned} I_{\lambda}(\tau \eta ) \leq {}& \tau ^{p_{1}^{-}}\sum _{i=1} ^{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (\mathcal{C} \Omega )^{2}}\frac{ \vert \eta (x)-\eta (y) \vert ^{{p_{i}(x,y)}}}{2{p_{i}}(x,y) \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy + \int _{\Omega} \frac{V_{i}(x) \vert \eta \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \\ &{} + \int _{\mathbb{R}^{N}\setminus \Omega} \frac{\beta (x) \vert \eta \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \biggr) - \lambda _{1}\tau ^{{r_{1}}^{-}+\epsilon _{1}} \int _{\Omega _{0}} \frac{A_{1}(x) \vert \eta \vert ^{r_{1}(x)}}{r_{1}(x)}\,dx \\ &{}-\lambda _{2}\tau ^{{r_{2}}^{-}+\epsilon _{2}} \int _{\Omega _{0}} \frac{A_{2}(x) \vert \eta \vert ^{r_{2}(x)}}{r_{2}(x)}\,dx - \int _{\mathbb{R}^{N} \setminus \Omega}\tau \eta g_{1}(x)\,dx - \int _{\mathbb{R}^{N} \setminus \Omega}\tau \eta g_{2}(x)\,dx \\ \leq {}&\frac{ \tau ^{p_{1}^{-}}}{p_{1}^{-}}3K_{1}\max \bigl\{ \Vert \eta \Vert _{X}^{p_{2}^{-}}, \Vert \eta \Vert _{X}^{p_{2}^{+}} \bigr\} - \frac{\lambda _{2}\tau ^{{r_{2}}^{-}+\epsilon _{2}}}{r_{2}^{+}} \int _{ \Omega _{0}}A_{2}(x) \vert \eta \vert _{X}^{r_{2}(x)}\,dx. \end{aligned}$$

Since ${r_{2}}^{-}+\epsilon _{2}< p_{1}^{-}$, we have $I_{\lambda}(t\eta )<0$.

In addition, combining the Hölder inequality and inequality (11), for any $\varphi \in B_{\rho} (0)$, we have

$$\begin{aligned} I_{\lambda}(\varphi ) &\geq \frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{1}^{-}} -\frac{2\lambda _{1}M_{1}^{r_{1}^{-}}}{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert _{X}^{r^{-}_{1}} - \frac{2\lambda _{2}M_{2}^{r_{2}^{-}}}{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert _{X}^{r^{-}_{2}} -2K_{1} \Vert \varphi \Vert _{X}. \end{aligned}$$

This fact gives

$$ -\infty < \widehat{c}:=\underline{\inf}_{B_{\rho}(0)}I_{\lambda}< 0. $$

(15)

Set

$$ \frac{1}{n}\in \bigl(0, \underline{\inf}_{\partial B_{\rho}(0)}I_{\lambda}( \varphi )-\underline{\inf}_{B_{\rho}(0)}I_{\lambda}(\varphi )\bigr),\quad n \in \mathbb{N}. $$

By (15), $I_{\lambda}:\overline{B_{\rho}(0)}\rightarrow \mathbb{R}$ is lower bounded on $\overline{B_{\rho}(0)}$ and $I_{\lambda}\in C^{1}(\overline{B_{\rho}(0)},\mathbb{R})$. Using Ekeland’s variational principle, there exists $\{\varphi _{n}\}_{n\in \mathbb{N}}\in \overline{B_{\rho}(0)}$ such that

$$ \textstyle\begin{cases} \widehat{c}\leq I_{\lambda}(\varphi _{n})\leq \widehat{c}+\frac{1}{n}, \\ 0< I_{\lambda}(\varphi _{n})- I_{\lambda}(\varphi ) +\frac{1}{n} \Vert \varphi _{n}-\varphi \Vert _{X}, \quad \varphi _{n}\neq \varphi . \end{cases} $$

(16)

Since

$$ I_{\lambda}(\varphi _{n})\leq \widehat{c}+\frac{1}{n} \leq \inf_{\overline{B_{\rho}(0)}}I_{\lambda}+\frac{1}{n} \leq \inf _{B_{\rho}(0)}I_{\lambda}+\frac{1}{n} < \inf _{ \partial B_{\rho}(0)}I_{\lambda}, $$

we have $\varphi _{n}\in B_{\rho}(0)$. Define function $\zeta : \overline{B_{\rho}(0)}\rightarrow \mathbb{R}$ by

$$ \zeta (\varphi )=I_{\lambda}(\varphi )+\frac{1}{n} \cdot \Vert \varphi _{n}-\varphi \Vert _{X}, $$

which implies $\zeta (\varphi _{n})<\zeta (\varphi )$ from (16). Then, $\varphi _{n}$ is a minimum point of ζ, and we have

$$ \frac{\zeta (\varphi _{n}+t\cdot v)-\zeta (\varphi _{n})}{t}\geq 0, $$

for small $t>0$ and any $v\in B_{1}(0)=\{v\in X: \|v\|_{X}=1\}$. Hence,

$$ \frac {I_{\lambda}(\varphi _{n}+t\cdot v)-I_{\lambda}(\varphi _{n})}{t}+ \frac{1}{n} \Vert v \Vert _{X}\geq 0. $$

Let $t\rightarrow 0$, then $\langle I'_{\lambda},v\rangle +\frac{1}{n}\|v\|_{X} \geq 0$. Replace v with −v. Then, we obtain $\langle -I'_{\lambda},v\rangle +\frac{1}{n}\|v\|_{X} \geq 0$. Thus, $\|I'_{\lambda}(\varphi _{n})\|_{X^{*}}\leq \frac{1}{n}$. We infer that there exists a sequence $\{\varphi _{n}\}_{n\in \mathbb{N}} \subset B_{\rho} (0)$ such that

$$ I_{\lambda}(\varphi _{n})\rightarrow \widehat{c}< 0, \qquad \bigl\Vert I_{\lambda}'( \varphi _{n}) \bigr\Vert _{X^{*}}\rightarrow 0,\quad n\rightarrow \infty . $$

By Lemma 4.2, there is $\varphi _{n}\rightarrow \varphi _{0}^{(2)}$ in X. Then, we have $I_{\lambda}'(\varphi _{0}^{(2)})=0$ and $I_{\lambda}(\varphi _{0}^{(2)})=\widehat{c}<0$, that is, $\varphi _{0}^{(2)}$ is another solution of equations (1) with negative energy, which ends the proof. □

Here, we give an example of application of Theorem 1.1.

Example 4.1

Let $\Omega =\{(x,y)\in \mathbb{R}: x^{2}+y^{2}\leq 1\}$. Consider the problem

$$ \textstyle\begin{cases} (-\Delta )^{\frac{1}{2}}_{x^{2}+y^{2}+3}\varphi +(-\Delta )^{ \frac{1}{2}}_{x^{2}+y^{2}+5}\varphi +2 \vert \varphi \vert ^{2x^{2}+1}\varphi +4 \vert \varphi \vert ^{2x^{2}+3}\varphi \\ \quad = \vert x \vert \vert \varphi \vert ^{\frac{1}{6}}\varphi +x^{2} \vert \varphi \vert ^{\frac{1}{2}} \varphi , \quad x \in \Omega , \\ \mathcal{N}_{\frac{1}{2},x^{2}+y^{2}+3}\varphi +\mathcal{N}_{ \frac{1}{2},x^{2}+y^{2}+5}\varphi +\ln \vert x \vert \vert \varphi \vert ^{2x^{2}+1} \varphi \\ \quad {}+\ln \vert x \vert \vert \varphi \vert ^{2x^{2}+3}\varphi =0,\quad x \in \mathbb{R}^{N} \setminus \overline{\Omega}. \end{cases} $$

(17)

By simple calculations, we obtain $\operatorname{meas}(\partial \Omega )=2\pi$, $p_{1}^{-}=3$, $p_{1}^{+}=4$, $p_{2}^{-}=5$, $p_{2}^{+}=6$. Conditions (P), (G), (H), and (V) are satisfied. We observe that all assumptions of Theorem 1.1 are fulfilled. Hence, Theorem 1.1 implies that problem (17) admits two nontrivial weak solutions.

5 Proof of Theorem 1.2

To prove Theorem 1.2, we first recall the following lemmas.

Lemma 5.1

([15]) Let X be a reflexive and separable Banach space. Then, there are $\{e_{n}\}\subset E$ and $\{e_{n}^{*}\}\subset E^{*}$ such that

$$ E=\overline{\operatorname{span}\{e_{n}: n=1,2,3\ldots \}},\qquad E^{*}= \overline{\operatorname{span}\bigl\{ e^{*}_{n}: n=1,2,3\ldots \bigr\} } $$

and

$$ \bigl\langle e_{i}^{*},e_{j} \bigr\rangle = \textstyle\begin{cases} 1, &\textit{if } i= j, \\ 0, &\textit{if } i\neq j. \end{cases} $$

Denote

$$ E_{n}=\operatorname{span}\{e_{n}\},\qquad X_{k}= \bigoplus_{n=1} ^{k}E_{n}, \quad \textit{and} \quad Y_{k}= \overline{\bigoplus_{n=k} ^{\infty}E_{n}}. $$

Lemma 5.2

([15]) Assume that $q(x)\in C_{+}(\overline{\Omega})$, $q(x)< p^{*}(x)$, for any $x\in \overline{\Omega}$ and denote

$$ \widetilde{\xi}_{k}=\sup_{\varphi \in Y_{k}, \Vert \varphi \Vert _{X}=1} \Vert \varphi \Vert _{L^{q(x)}(\Omega )}, $$

then $\lim_{k\rightarrow \infty}\widetilde{\xi}_{k}=0$.

Now, we recall the fountain theorem.

Theorem 5.1

([15]) Let X be a real Banach space and $I_{\lambda}\in C^{1}(X, \mathbb{R})$ be a even functional satisfying the $(PS)$ conditions. There exists $r_{k}>0$ such that $\widehat{\rho}_{k}> r_{k}>0$ for every $k\in \mathbb{N}$. Then, the following conditions hold:

• (i) $\alpha _{k}=\max \{I_{\lambda}(\varphi ): \varphi \in X_{k}, \| \varphi \|=\widehat{\rho}_{k}\}\leq 0$;

• (ii) $\beta _{k}=\inf \{I_{\lambda}(\varphi ): \varphi \in Y_{k}, \| \varphi \|= r_{k}\}\rightarrow +\infty$ as $k\rightarrow \infty$.

Then, $I_{\lambda}$ possesses a series of critical points $\varphi _{k}$ such that $I_{\lambda}(\varphi _{k})\rightarrow +\infty$.

Lemma 5.3

Assume that assumptions (P), (G), (V), and (H) hold. There exists $\widehat{\rho}_{k}>0$ such that

$$ \max_{\varphi \in X_{k}, \|\varphi \|=\widehat{\rho}_{k}}I_{\lambda}( \varphi )< 0. $$

Proof

Let $t\in (0,1)$. For $\|\widehat{\varphi}\|_{X}=\widetilde{\rho}_{k}\geq 1$ and $\widehat{\rho}_{k}>\widetilde{\rho}_{k}$, there exists φ̂ such that

$$\begin{aligned} I_{\lambda}(t\widehat{\varphi}) \leq{} & t^{p_{2}^{+}}\sum _{i=1} ^{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (\mathcal{C} \Omega )^{2}}\frac{ \vert \widehat{\varphi}(x)-\widehat{\varphi}(y) \vert ^{{p_{i}(x,y)}}}{2{p_{i}}(x,y) \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy + \int _{\Omega}V_{i}(x) \frac{ \vert \widehat{\varphi} \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \\ &{} + \int _{\mathbb{R}^{N}\setminus \Omega} \frac{\beta (x) \vert \widehat{\varphi} \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \biggr) -\lambda _{1}t^{{r_{1}}^{-}} \int _{\Omega} \frac{A_{1}(x) \vert \widehat{\varphi} \vert ^{r_{1}(x)}}{r_{1}(x)}\,dx \\ &{}-\lambda _{2}t^{{r_{2}}^{-}} \int _{\Omega} \frac{A_{2}(x) \vert \widehat{\varphi} \vert ^{r_{2}(x)}}{r_{2}(x)}\,dx - \int _{ \mathbb{R}^{N}\setminus \Omega}t\widehat{\varphi} g_{1}(x)\,dx - \int _{ \mathbb{R}^{N}\setminus \Omega}t\widehat{\varphi} g_{2}(x)\,dx \\ \leq {}&\frac{ t^{p_{2}^{+}}}{p_{1}^{-}} \Vert \widehat{\varphi} \Vert _{X}^{p_{2}^{+}} -\frac{\lambda _{1}t^{{r_{1}}^{-}}}{{r_{1}^{+}}} \int _{\Omega}A_{1}(x) \vert \widehat{\varphi} \vert ^{r_{1}(x)}\,dx < 0 \end{aligned}$$

with $p_{1}^{-}>r_{1}^{+}>1$. Taking $\varphi =t\widehat{\varphi}$, for sufficiently small t, it follows that

$$ \alpha _{k}=\max_{\varphi \in X_{k}, \|\varphi \|=\widehat{\rho}_{k}} I_{\lambda}(\varphi )\leq 0. $$

 □

Lemma 5.4

Assume that assumptions (P), (G), (V), and (H) hold. There exists $r_{k}>0$ such that

$$ \inf_{\varphi \in Y_{k}, \|\varphi \|=r_{k}}I_{\lambda}(\varphi )>+ \infty . $$

Proof

According to Lemma 5.2, for $\|\varphi \|_{X}=r_{k}>1$, we obtain

$$\begin{aligned} I_{\lambda}(\varphi ) \geq {}&\frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{1}^{-}} -\frac{\lambda _{1}}{r_{1}^{-}} \int _{\Omega}A_{1}(x) \vert \varphi \vert ^{r_{1}(x)}\,dx -\frac{\lambda _{2}}{r_{2}^{-}} \int _{\Omega}A_{2}(x) \vert \varphi \vert ^{r_{2}(x)}\,dx \\ \geq {}&\frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{1}^{-}} - \frac{2\lambda _{1}M_{1}^{r_{1}^{+}}}{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert _{X}^{r_{1}^{+}} - \frac{2\lambda _{2}M_{2}^{r_{2}^{+}}}{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert _{X}^{r_{2}^{+}} \\ \geq{} &\frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{1}^{-}} - \frac{2\lambda _{1}M_{1}^{r_{1}^{+}}\widetilde{\xi}_{k}^{r_{1}^{+}}}{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert _{X}^{r_{1}^{+}} - \frac{2\lambda _{2}M_{2}^{r_{2}^{+}}\widetilde{\xi}_{k}^{r_{2}^{+}}}{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert _{X}^{r_{2}^{+}}. \end{aligned}$$

Let

$$\begin{aligned} &\max \biggl\{ \frac{2\lambda _{1}M_{1}^{r_{1}^{+}}\widetilde{\xi}_{k}^{r_{1}^{+}}}{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert _{X}^{r^{+}_{1}}, \frac{2\lambda _{2}M_{2}^{r_{2}^{+}}\widetilde{\xi}_{k}^{r_{2}^{+}}}{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert _{X}^{r^{+}_{2}} \biggr\} \\ &\quad =2 \frac{\lambda _{\varphi}M_{\varphi}^{r_{\varphi}}\widetilde{\xi}_{k}^{\varphi}}{r_{\varphi}^{-}} A_{\varphi}^{*} \Vert \varphi \Vert _{X}^{r_{\varphi}}, \end{aligned}$$

and there exists a constant C̃ such that $r_{\varphi}=\max \{r_{1}^{+}+\widetilde{C}, r_{2}^{+}+\widetilde{C} \}$, where $1< p_{1}^{-}< r_{\varphi}^{-}< r_{\varphi}$. Therefore,

$$\begin{aligned} &\frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{1}^{-}} - \frac{2\lambda _{1}M_{1}^{r_{1}^{+}}\widetilde{\xi}_{k}^{r_{1}^{+}}}{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert _{X}^{r_{1}^{+}} - \frac{2\lambda _{2}M_{2}^{r_{2}^{+}}\widetilde{\xi}_{k}^{r_{2}^{+}}}{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert _{X}^{r_{2}^{+}} \\ &\quad \geq \Vert \varphi \Vert _{X}^{p_{1}^{-}} \biggl( \frac{1}{p_{2}^{+}} -4 \frac{\lambda _{\varphi}M_{\varphi}^{r_{\varphi}}\widetilde{\xi}_{k}^{\varphi}}{r_{\varphi}^{-}} A_{\varphi}^{*} \Vert \varphi \Vert _{X}^{r_{\varphi}-p_{1}^{-}} \biggr). \end{aligned}$$

Choose

$$ r_{k}= \biggl( \frac{5\lambda _{\varphi}M_{\varphi}^{r_{\varphi}}p_{2}^{+}\widetilde{\xi}_{k}^{\varphi}}{r_{\varphi}^{-}} A_{\varphi}^{*} \biggr)^{\frac{1}{p_{1}^{-}-r_{\varphi}}}. $$

Since $p_{1}^{-}< r_{\varphi}$, we have $r_{k}\rightarrow +\infty$ as $k\rightarrow +\infty$. By the choice of $r_{k}$ with $\|\varphi \|_{X}=r_{k}$ such that $\widehat{\rho}_{k}>\widetilde{\rho}_{k}>r_{k}>0$, we obtain

$$ \beta _{k}=\inf_{\varphi \in Y_{k}, \|\varphi \|=r_{k}}I_{\lambda}( \varphi ) \rightarrow +\infty ,\quad k\rightarrow +\infty . $$

 □

Proof of Theorem 1.2

Let hypotheses (P), (G), (V), and (H) be satisfied. By Lemma 4.2, $I_{\lambda}$ satisfies $(PS)$ conditions. Under the definition of $I_{\lambda}$ in Lemma 3.13, it follows that $I_{\lambda}(0)=0$ and $I_{\lambda}$ is an even function. Therefore, from Lemmas 5.3 and 5.4, it can be deduced that $I_{\lambda}$ satisfies Theorem 5.1. Then, $I_{\lambda}$ possesses a series of critical points $\varphi _{k}$ as $k\rightarrow +\infty$. In conclusion, equations (1) possess infinitely many nontrivial weak solutions. □

Here, we give an example of application of Theorem 1.2.

Example 5.1

Let $\Omega =\{(x,y)\in \mathbb{R}: x^{2}+y^{2}\leq 1\}$. Consider the problem

$$ \textstyle\begin{cases} (-\Delta )^{\frac{1}{2}}_{x^{2}+y^{2}+3}\varphi +(-\Delta )^{ \frac{1}{2}}_{x^{2}+y^{2}+5}\varphi +(x^{2}+1) \vert \varphi \vert ^{2x^{2}+1} \varphi +(e^{ \vert x \vert }+2) \vert \varphi \vert ^{2x^{2}+3}\varphi \\ \quad = \vert x \vert \vert \varphi \vert ^{\sqrt{ \vert x \vert }-\frac{3}{4}}\varphi +x^{2} \vert \varphi \vert ^{ \sqrt{ \vert x \vert }-\frac{1}{4}}\varphi , \quad x \in \Omega , \\ \mathcal{N}_{\frac{1}{2},x^{2}+y^{2}+3}\varphi +\mathcal{N}_{ \frac{1}{2},x^{2}+y^{2}+5}\varphi +\ln \vert x \vert \vert \varphi \vert ^{2x^{2}+1} \varphi \\ \quad {}+\ln \vert x \vert \vert \varphi \vert ^{2x^{2}+3}\varphi =0, \quad x \in \mathbb{R}^{N} \setminus \overline{\Omega}. \end{cases} $$

(18)

By simple calculations, we obtain $\operatorname{meas}(\partial \Omega )=2\pi$, $p_{1}^{-}=3$, $p_{1}^{+}=4$, $p_{2}^{-}=5$, $p_{2}^{+}=6$, $r_{1}^{-}=5/4$, $r_{1}^{+}=9/4$, $r_{2}^{-}=7/4$, and $r_{2}^{+}=11/4$. That is, conditions (P), (G), (H), and (V) are satisfied. We observe that all assumptions of Theorem 1.2 are fulfilled. Hence, Theorem 1.2 implies that problem (18) admits infinitely many nontrivial weak solutions.

6 Proof of Theorem 1.3

We give some results with the aid of the Krasnoselskii genus. Let E be a real Banach space and set

$$ \mathfrak{R}=\bigl\{ \mathcal{A}\subset E\setminus \{0\}: \mathcal{A} \text{ is compact and } \mathcal{A}=-\mathcal{A}\bigr\} . $$

Let $\mathcal{A}\subset \mathfrak{R}$ and $E=\mathbb{R}^{k}$. We define genus

$$ \gamma (\mathcal{A})=\min \bigl\{ k\geq 1: \text{there exists an odd continuous mapping } \psi :\mathcal{A} \rightarrow \mathbb{R}^{k}\setminus \{0\} \bigr\} . $$

If the mapping ψ does not exist for any $k>0$, and set $\gamma (\mathcal{A})=\infty$. If $\mathcal{A}$ is a subset consisting of a finite number of pairs of points, then, $\gamma (\mathcal{A})=1$. Furthermore, from definition, ${\gamma (\emptyset )=0}$.

Lemma 6.1

([18]) Let $E=R^{N}$ and ∂Ω be the boundary of an open, symmetric, and bounded subset $\partial \Omega \subset R^{N}$ with $0\in \Omega$. Then, $\gamma (\partial \Omega )=N$.

Corollary 6.1

([18]) $\gamma (S^{N-1})=N$.

Theorem 6.1

([18]) Let $I_{\lambda}\in C^{1}(X)$ be a functional satisfying the $(PS)$ conditions and assume that

• (i) $I_{\lambda}$ is bounded from below and even;

• (ii) there is a compact set $K\in \mathbb{R}$ such that $\gamma (K)=k$ and $\sup_{x\in K}I_{\lambda}(x)< I_{\lambda}(0)$.

Then, $I_{\lambda}$ has at least k pairs of distinct critical points whose corresponding critical values are all less than $I_{\lambda}(0)$.

Proof of Theorem 1.3

Combining (12), (13), and the Hölder inequality (7) for $\|\varphi \|_{X}>1$, we obtain

$$\begin{aligned} I_{\lambda}(\varphi ) \geq {}&\frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{1}^{-}} -\frac{\lambda _{1}}{r_{1}^{-}} \Vert A_{1} \Vert _{s_{1}(x)} \Vert \varphi \Vert _{L^{s'_{1}(x)r_{1}(x)}( \Omega )}^{r^{+}_{1}} - \frac{\lambda _{2}}{r_{2}^{-}} \Vert A_{2} \Vert _{s_{2}(x)} \Vert \varphi \Vert _{L^{s'_{2}(x)r_{2}(x)}(\Omega )}^{r^{+}_{2}} \\ \geq {}& \frac{1}{p_{2}^{+}} \Vert \varphi \Vert _{X}^{p_{1}^{-}} - \frac{\lambda _{1}}{r_{1}^{-}}2 \Vert A_{1} \Vert _{s_{1}(x)} M_{1}^{r_{1}^{+}} \Vert \varphi \Vert _{X}^{r^{+}_{1}} -\frac{\lambda _{2}}{r_{2}^{-}}2 \Vert A_{2} \Vert _{s_{2}(x)} M_{2}^{r_{2}^{+}} \Vert \varphi \Vert _{X}^{r^{+}_{2}}. \end{aligned}$$

Since $\max \{1, r^{+}_{1}, r^{+}_{2}\}< p_{1}^{-}$, for $\|\varphi \|_{X}$ large enough, $I_{\lambda}$ is bounded from below. $I_{\lambda}\in X$ is an even function by the definition and $I_{\lambda}(0)=0$. Moreover, $I_{\lambda}$ is coercive in X and satisfies the $(PS)$ conditions by Lemma 4.2. Let

$$ \mathfrak{R}_{k}=\bigl\{ \mathcal{M}\subset \mathfrak{R}:\gamma ( \mathcal{M})\geq k\bigr\} , \qquad c_{k}=\inf_{\mathcal{M}\in \mathfrak{R}_{k}} \sup _{\varphi \in \mathcal{M}}I(\varphi ),\quad k=1,2,\ldots . $$

We obtain

$$ -\infty < c_{1}\leq c_{2}\leq \cdots \leq c_{k} \leq c_{k+1} \leq \cdot \cdot \cdot . $$

Now we prove that for any $k \in \mathbb{N}$, there is $c_{k}<0$. For each k, we take k disjoint open sets $\widetilde{K}_{i}$ such that $\bigcup^{k}_{i=1} \widetilde{K}_{i}\subset \Omega$. For $i = 1,\ldots ,k$, let $\varphi _{i} \in (X\bigcap C^{\infty}_{0}(\widetilde{K} _{i}) \setminus \{0\}$ with $\|\varphi _{i}\|_{X} = 1$, and

$$ \mathcal{M}_{k}=\operatorname{span} \{\varphi _{1},\varphi _{2},\ldots, \varphi _{k}\}. $$

Since each norm on $\mathcal{M}_{k}$ is equivalent, there is $\rho ^{*}_{k} \in (0,1)$ such that $\varphi \in \mathcal{M}_{k}$ with $\|\varphi \|_{X} \leq \rho ^{*}_{k}$, which means that $\|\varphi \|_{\infty}< C_{\rho ^{*}_{k}}<1$. Set

$$ S^{(k)}_{\rho ^{*}_{k}}=\bigl\{ \varphi \in \mathcal{M}_{k}: \Vert \varphi \Vert _{X} = \rho ^{*}_{k} \bigr\} . $$

Combining the compactness of $S^{(k)}_{\rho ^{*}_{k}}$ and $t\in (0,1)$ for all $\varphi \in S^{(k)}_{\rho ^{*}_{k}}$,

$$\begin{aligned} I_{\lambda}(t\varphi ) \leq {}& t^{p_{1}^{-}}\sum _{i=1} ^{2} \biggl( \int _{\mathbb{R}^{2N}\setminus (\mathcal{C} \Omega )^{2}}\frac{ \vert \varphi (x)-\varphi (y) \vert ^{{p_{i}(x,y)}}}{2{p}_{i}(x,y) \vert x-y \vert ^{N+sp_{i}(x,y)}}\,dx\,dy + \int _{\Omega}V_{i}(x) \frac{ \vert \varphi \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \\ &{} + \int _{\mathbb{R}^{N}\setminus \Omega} \frac{\beta (x) \vert \varphi \vert ^{\bar{p}_{i}(x)}}{\bar{p}_{i}(x)}\,dx \biggr) - \int _{\mathbb{R}^{N}\setminus \Omega}t\varphi g_{1}(x)\,dx - \int _{ \mathbb{R}^{N}\setminus \Omega}t\varphi g_{2}(x)\,dx \\ &{}-\lambda _{1}t^{r_{1}^{-}} \int _{\Omega} \frac{A_{1}(x) \vert \varphi \vert ^{r_{1}(x)}}{r_{1}(x)}\,dx -\lambda _{2}t^{r_{2}^{-}} \int _{\Omega} \frac{A_{2}(x) \vert \varphi \vert ^{r_{2}(x)}}{r_{2}(x)}\,dx \\ \leq{} &3K_{1}t^{p_{1}^{-}} \Vert \varphi \Vert ^{p_{2}^{+}}_{X}- \frac{\lambda _{1}t^{r_{1}^{-}}}{p_{1}^{-}} \int _{\Omega}A_{1}(x) \vert \varphi \vert ^{r_{1}(x)}\,dx. \end{aligned}$$

Sine $1< r_{1}^{-}< p_{1}^{-}$, there exist $t_{k}\in (0,1)$ and $\varepsilon _{k}$ such that

$$ I_{\lambda}(t_{k}\varphi )< -\varepsilon _{k}< 0. $$

Thus, $I_{\lambda}(\varphi )<0$ for all $\varphi \in S^{(k)}_{t_{k}\rho ^{*}_{k}}$. Furthermore, $\gamma (S^{(k)}_{t_{k}\rho ^{*}_{k}})=k$ such that $c_{k}<-\varepsilon _{k}<0$ for all k, and the assertion is proved. Each $c_{k}$ is a critical value by the Krasnoselskii genus theory. Combining Theorem 6.1, $I_{\lambda}$ has at least k pairs of different critical points. In addition, since k is arbitrary, we obtain an infinite number of critical points of equations (1). □